the pension challenge · 2019. 12. 12. · “chap09” — 2003/6/4 — page 187 — #1 chapter 9...

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The PensionChallenge

Risk Transfers andRetirement IncomeSecurityEDITED BY

Oliva S. Mitchell and Kent Smetters

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Chapter 9

Money-Back Guarantees in IndividualPension Accounts: Evidence fromthe German Pension Reform

Raimond Maurer and Christian Schlag

The German Retirement Saving Act (‘‘Altersvermögensgesetz’’1) whichpassed the German legislative body in May of 2001 instituted a new fundedsystem of supplementary pensions coupled with a general reduction in thelevel of state pay-as-you-age pensions. The goal of this new pension system isto cap and to stabilize the contributions of German employees to the statepension system, which cost 19.1 percent of salary. For compulsory membersof the state pension systems not already in retirement, the maximum ‘‘firstpillar’’ state pension level will be gradually cut from 70 to 67 percent of thelast net salary before retirement by 2030.2 To compensate for the cut instate pension payouts, individuals will be able to invest voluntarily and on apre-tax basis a part of their income in individual pension accounts (‘‘Alters-vorsorgevertrag,’’ called here IPAs).3 Additional incentives to invest into theIPAs are given by the government in the form of a tax relief on pension con-tributions, direct subsidies for low income earners, and extra contributionsfor children. In order to get the full benefits, households will have to investabout 1 percent of their income (up to the social security ceiling) into thepension system in 2002, increasing every 2 years by 1 percent reaching amaximum of 4 percent in 2008. The investment income during the accu-mulation period is not subject to income tax, whereas the payments fromthe IPA during the distribution phase will be fully subject to income tax.

In general, individuals are free to make IPA pension investments in awide array of products offered by private sector financial institutions. Thisallows participants to choose an investment portfolio that is consistent withtheir individual preferences for risk and return. In order to qualify for atax credit, however, the IPA products have to satisfy a number of criteria.

This research is part of the Research Program ‘‘Institutional Investors’’ of the Center for Fin-ancial Studies, Frankfurt/M. This chapter was written in part during Dr. Maurer’s time asthe Metzler Visiting Professor at the Wharton’s School Pension Research Council. The authorswould like to thank Manfred Laux, David McCarthy, Olivia S. Mitchell, Alex Muermann, RudolfSiebel, Wolfgang Raab, and Kent Smetters for helpful comments. Opinions and errors are solelythose of the authors.

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188 Maurer and Schlag

These conditions are codified in a special law concerning the certificationof individual pension products (‘‘Altersvorsorge-Zertifizierungsgesetz’’) andsupervised by a special authority (‘‘Zertifizierungsstelle’’) belonging tothe German Federal Financial Supervisory Agency. The intention of thecertification requirements is twofold:

1. First, the government wants to ensure that individuals only use the(tax-supported) accumulated savings for a lifelong income streamin the post-retirement phase, and not for consumption during pre-retirement.

2. Second, private (and often uninformed) investors paying into the newindividual pension plans should be protected against the risk of making‘‘too bad’’ investment decisions.

In the spirit of the first intention, investments in the personal pensionaccounts must be preserved until employees reach the age of 60,4 andno distributions may be made during the accumulation period. When theage of retirement is reached, the accumulated assets must be drawn downin the form of a lifelong annuity or a capital withdrawal plan which must(partly) revert into an annuity at the age of 85.5 To provide transparency, theproviders of IPAs must disclose the nature and level of fees (e.g. to cover dis-tribution and/or administrative costs). If distribution fees are not chargedas a percentage of the periodic contribution into the plan, they must bespread equally over a period of at least 10 years.6 During the accumulationphase, the policyholder has the right to suspend the contract as well as toterminate the contract by switching the cash value of the policy to a newprovider.

In line with a certain minimum level of investor protection, only regulatedfinancial institutions, like banks, life insurance companies, and mutual fundcompanies, are allowed to offer IPAs. In principle, these providers are free todesign their IPA. In particular, the Certification Act imposes no restrictionsconcerning the assets in which the providers invest the contribution thatback the pension accounts.7 In addition, the supervisory authority does notcheck whether the risk and return characteristics of an IPA are ‘‘economic-ally feasible.’’ Yet, the provider of an IPA must promise the plan participantthat the contract cash value at retirement is at least equal to contributionsmade to the IPAs, including all extra payments by the government.8 This‘‘money-back’’ guarantee, which was the core of an intense and controver-sial debate during the social security reform in Germany, is the focus of thischapter. Advocates of the guarantee argue that it protects plan participantsagainst a portion of the downside volatility of capital market returns, byproviding them with a minimum rate of return with respect to their lifetimecontributions. However, the guarantee shapes the design of saving products

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9 / Money-Back Guarantees 189

offered by the providers and raises a question about the economic costs ofsuch a promise.9 Depending on the assets used to back the pension accounts,providers may be exposed to shortfall risk due to adverse movements in cap-ital markets. If, at retirement, the value of the pension assets is lower thanthe sum of the contributions paid into the plan, the IPA provider must fillthe gap with its own equity capital. The problem faced by money managersis therefore to find a product design, in conjunction with an appropriateinvestment strategy, that protects the credibility of the guarantee in scenariosof negative investment returns (hedging effectiveness), while still allowingfor sufficient upside potential if capital markets are booming (and thusavoiding excessive hedging costs). In addition, the guarantee has importantimplications for regulators who must find an effective and efficient solvencysystem for such saving schemes, especially for mutual funds.

The objective of this chapter is to explore how this money-back guaranteeworks for products offered by the German mutual fund industry. We evalu-ate alternative designs for guarantee structures including a life cycle model(dynamic asset allocation), a plan with a pre-specified blend of equity andbond investments (static asset allocation), and some type of portfolio insur-ance. We use simulation to compare hedging effectiveness and hedgingcosts associated with the provision of the money-back guarantee.

Long-Term and Shortfall Risks, andReturn of Saving PlansIn order to make appropriate investment decisions under uncertainty, indi-viduals must be able to compare the risk and rewards of different assetclasses. Yet policymakers, regulators, and providers are also interested inthe long run performance of financial assets that back the new pensionproducts. The impact of the investment horizon on the risk of the variousfinancial assets is still a subject of intense and controversial debate within theacademic community and among investment professionals.10 For example,a popular statement is that stocks have a lower downside side risk in the longrun than in the short run. A practical guideline based on this argument isthat people should invest a higher fraction of their money in stocks theyounger they are, independent of preferences.11 If the time horizon is longenough, this approach would imply that people should invest 100 percent instocks. To justify this view, proponents call on the law of large numbers which(seemingly) forces a phenomenon called ‘‘time diversification.’’ Intuitively,this means that over a sufficiently long investment horizon, losses resultingfrom the high downside fluctuations will be compensated by gains result-ing from the high upside fluctuations of short term stock returns. Someinvestment advisors press this argument by pointing to historical returnsand demonstrating that stocks have outperformed bonds for every 10-, 15-,or 20-year period on record.

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Nevertheless, it is well known in the academic literature that this is amisleading argument. For example, Samuelson (1963) uses utility theory,Levy and Cohen (1998) use stochastic dominance, and Bodie (1995, 2001)applies option pricing theory to demonstrate the logical flaw in this conjec-ture. In addition, the use of historical return series implies that the 10-, 15-,or 20-year periods used are strongly overlapping, so the resulting rollovermultiperiod returns have a high degree of correlation, both of which resultin a serious estimation bias.

Shortfall Risk MeasuresThis section provides additional evidence concerning the impact of the timehorizon on the risk of the major financial asset classes in the German context,that is, stocks and bonds. To do so, we use alternative shortfall risk measures.The concept of shortfall risk is associated with the possibility of ‘‘somethingbad happening,’’ in other words, falling short compared to a required target(benchmark) return.12 Returns below the target (losses) are considered tobe undesirable or risky, while returns above the target (gains) are desirableor non-risky. In this sense, shortfall risk measures are called ‘‘relative’’ or‘‘pure’’ measures of risk.

A popular measure to examine the downside risk of different investmentvehicles is the shortfall probability. Formally, let R denote the cumulative(multiyear) return of an investment at a specific point in time. Then theshortfall probability is given by

SP = Prob(R < z), (9.1)where z is the target (benchmark) which translates the total investmentreturns into gains or losses. In the special case of a money back guarantee,the target is set equal to zero; that is, the shortfall occurs when the cash valueof the policy is lower than the premiums paid into the saving plan. Despitethe popularity of this risk measure in the investment industry, it has a majorshortcoming. As Bodie (2001: 308) points out it ‘‘completely ignores howlarge the potential shortfall might be.’’ If the same investment strategy canbe repeated many times, the shortfall probability only answers the question‘‘how often’’ a loss might occur, but not ‘‘how bad’’ such a loss might be.

To provide information about the potential extent of a loss, we calculatethe Mean Excess Loss (MEL), also known as the conditional shortfall expecta-tion, as an additional measure to evaluate the long-term and shortfall risk offinancial assets. Formally, this risk index is given by

MEL = E[z − R |R < z], (9.2)and it indicates the expected loss with respect to the benchmark, under thecondition that a shortfall occurs. Therefore, given a loss, the MEL answers

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the question ‘‘how bad on average’’ the loss will be.13 In this sense, the MELcan be considered a worst case risk measure, since the measure only con-siders the consequences of the mean shortfall-level assuming that a shortfallhappens.

A shortfall risk measure which connects the probability and the extent ofthe conditional shortfall in an intuitive way is the shortfall expectation (SE):

SE = E[max(z − R , 0)] = SP × MEL (9.3)The shortfall expectation is the sum of losses weighted by their probab-ilities, and hence it is a measure of the unconditional ‘‘average loss.’’ Asequation (9.3) shows, the mean shortfall level is simply the product of theshortfall probability and the mean level of shortfall, given the occurrence ofa shortfall. In addition, the SE is, in a certain way, related to the price of aninsurance contract which would cover the shortfall. For example, the pro-vider may have the possibility of transferring the shortfall risk to the capitalmarket by using appropriate arbitrage-free put options. Then the shortfallexpectation between the cash value of the pension assets and the guaranteepayment with respect to the risk adjusted (‘‘martingale’’) probabilities dis-counted back at the risk-free interest rate, results in the (modified) Blackand Scholes (1973) option pricing formula.14 If the provider transfers the AQ: Black

and Scholes(1973) notlisted.

shortfall risk to a reinsurance company, the shortfall expectation could beseen as an important element of an appropriate premium.

CalibrationNext we quantify and compare the shortfall risk (in the sense defined above)with respect to the preservation of principal of two saving plans. The firstinvests the contribution into stock index fund units, represented by theGerman stock index (DAX). The other saving plan is based on bond indexfund units, represented by the German bond index (REXP). The DAXstands for an index portfolio of German blue chips, and the REXP representsportfolio of German government bonds. Each of these indices is adjustedfor capital gains as well as dividends and coupon payments (on a pre-taxbasis). We assume a series of equal contributions paid at the beginning ofeach month up to the end of the accumulation period, which ranges from1 to 20 years.

To gain information about the relevant risk measures, we employ an ex anteapproach by imposing an exogenous structure on the probability distribu-tion governing the uncertainty of future asset returns. With such a model, itis possible to look into the future and compute the risk measures in which weare interested. Due to the complexity of the underlying payment structureof saving plans, there are no analytical closed form expressions for theserisk-measures.15 Therefore we use Monte-Carlo simulation to generate alarge number of paths for the evolution of the saving plans.

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The relevant statistics for the shortfall risk measures are then evaluated onthe basis of these scenarios. The stochastic dynamics of the (uncertain) mar-ket values of investment fund units are posited to follow geometric Brownianmotion, a standard assumption in financial economics that may be tracedback to Bachelier (1900). This implies that the log-returns of each type of

AQ:Bachelier(1900) innot listed

index fund are independent identically and normally distributed. For theestimation of the process parameters (drift/diffusion), we use the histor-ical monthly log-returns of the DAX and REXP over the period January,1973--December, 2001. The mean log rates of return for stocks (bonds) are0.7967 percent per month (0.5683 percent p.m.) and the correspondingstandard deviation 5.58 percent p.m. (1.12 percent p.m.). To take potentialadministration costs into account, we subtract the equivalent of 0.5 percentper annum (p.a.) from the monthly average return on the investments.16

Compatible with the prevalent German mutual fund fee structure, we takemarketing costs into consideration by assuming front end sales charges of5 percent for the stock and 3 percent for the bond fund units.

With respect to these parameters and consistent with the model of a geo-metric Brownian motion, we generated 3,000,000 random paths for thedevelopment of the pension plan with an investment horizon of 20 years(240 months).17 For each simulation path i(i = 1, . . . , 3,000,000) we com-pute for each month t(t = 1, . . . , 240) the (uncertain) compounded(multiyear) return, that is,

Ri ,t = Vi ,t − PtPt . (9.4)

Here Vi ,t stands for the cash value of the IPA in month t(t = 1, . . . , T ) insimulation run i(i = 1, . . . , n) and Pt for the sum of contributions paid untilmonth t . According to the money-back guarantee, we set the benchmarkreturn equal z = 0. With respect to this target, the relevant risk paramet-ers are then determined on the basis of the spectrum of possible futuredevelopments.

ResultsWe start with the results for the development of the expected (multiyear)return and the shortfall probability of the stock and bond index fund overtime. The graphs in Figure 9-1 indicate that a German investor would havethe potential to receive a substantially higher expected return by investingin stocks instead of bonds. For example, in the case of stocks at the end of a20-year accumulation period, the investor can expect a compounded returnwith respect to his contributions of 270 percent. For a saving plan basedon bond index funds, the expected return is only 109 percent. However,purchasing such an investment exposes the plan participant to the volatilityand therefore the downside risk of financial markets.

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0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20

Exp

ecte

d C

ompo

unde

d R

etur

n (%

of c

ontr

ibut

ions

)

Bonds

Time (years)

Stocks

Figure 9-1. Expected compounded return of saving plans in stocks and bonds.(Source : Authors’ Computations.)

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18 20

Bonds

Stocks

Time (years)

Sho

rtfa

ll P

roba

bilit

y(in

%)

Figure 9-2. Shortfall probability against a (nominal) zero percent target rate of returnin stock and bond saving plans (Source : Authors’ Computations.)

Next we illustrate the results for the development of the shortfall probab-ility of a saving plan using stock and bond index funds over time. Figure 9-2shows the well-known effect of time diversification, implying that the risk ofnot maintaining nominal capital decreases monotonically with an increasinginvestment period for bonds and stocks. Yet the rate and the extent of therisk reduction differ notably between the two investment vehicles. For bondindex funds, the shortfall probability is 37 percent for a yearly investment,

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0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16 18 20

Time (years)

Bonds

StocksME

L(%

of c

ontr

ibut

ions

)

Figure 9-3. Conditional mean expected loss (MEL) against a (nominal) zeropercent target rate of return in stock and bond saving plans. (Source : Authors’Computations.)

and close to zero (i.e. lower than 0.1 percent) with an investment horizon of7 years onwards. By contrast, the shortfall probability of a stock index funddoes not converge as rapidly towards zero. Thus, even for longer time hori-zons, the shortfall probability remains at a substantial level. For example,the shortfall probability for a 12-month saving plan is 48.09 percent, andfor accumulation period of 20 years it is still 2.72 percent. In principle,these results confirm a characteristic which Leibowitz and Krasker (1988)call persistence of risk.

Corresponding results for the MEL are presented in Figure 9-3. Savingplans in stocks have an MEL that increases monotonically with the lengthof the accumulation period, in contrast to bonds. For example, for an accu-mulation period of 1 year (i.e. 12 months) the conditional expected lossis 8.62 percent of the sum of the contribution paid into the stock pen-sion plan, while for a holding period of 20 years (i.e. 240 months) thisrisk index increases to 16.53 percent. For a pension plan using bond indexfunds, the MEL is 1.63 percent after 1 year, while for accumulation periodsof 13 years onwards, none of the 3,000,000 simulation paths produces ashortfall. Hence, with respect to the magnitude of a potential shortfall, thepopular argument that stocks become less risky in the long run is not true.

This result is in line with Samuelson’s (1963) finding concerning the fal-lacy of the law of large numbers. In addition, these results make clear thatthe use of the shortfall probability alone is a misleading risk measure of stockinvestments in the long run. The worst-case aspect of a long-term investmentin stocks is partly hidden by only taking the shortfall probability into consid-eration. Bodie (2002) provided the following very intuitive explanation forthis result ‘‘the probability of a bad thing happening is only part of the risk

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0

1

2

3

4

5

6

Sho

rtfa

ll E

xpec

tatio

n(in

% o

f con

trib

utio

ns)

Bonds

Stocks

Time (years)0 2 4 6 8 10 12 14 16 18 20

Figure 9-4. Expected shortfall against a (nominal) zero percent target rate of returnin stock and bond saving plans (Source : Authors’ Computations.)

equation. The other part is the severity of that bad thing, and the further outyou go, the more severe it could be.’’ Thus, the elucidation of the worst-caserisk embodied in a long-term investment in stocks represents an additionalpiece of information that might be essential for investors.

Figure 9-4 shows how the unconditional shortfall expectation developsover time. For a saving plan in bond index funds, both the probability ofloss and the mean excess loss decrease with the length of the time horizon.Because the shortfall expectation measures the net effect of both risk com-ponents, it is also decreasing in time. For a stock-based saving plan, this riskmeasure is also decreasing, that is the decreasing shortfall probability over-compensates the increasing MEL, to a certain extent. However, in contrastto bonds, we can observe a risk persistence-characteristic in the stock fund:even for very long time horizons, the shortfall expectation remains at asubstantial level.

In summary, even for long investment horizons, a pure stock investmentis not free of the downside risk of losing money. Hence it is not possibleto perfectly smoothen the negative short-run fluctuations of stock returnsover long horizons and simultaneously, to keep expected excess returns withcertainty. Consequently, assets with low volatility and low expected returns,like bonds, are not superfluous in the design of long-term saving products.Insurance contracts covering the shortfall of a principal guarantees are notcostless for a pure stock investment, even for long investment horizons (seeLachance and Mitchell, Chapter 8, this volume).

For low volatility assets like a portfolio of government bonds, the prob-ability and the severity of losing money decreases over time. Over longinvestment horizons, the price to insure the downside risk of a principalguarantee for a pure bonds investment is very low (close to zero). Hence

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bond pension plans are very effective vehicles for producing principal guar-antees. Of course, this does not mean that with a pure bond pension plan,the economic costs of downside protection is zero. Providers of bond-basedIPA’s must give up a substantial part of the upside returns that are possiblewith stocks. From an ex ante point of view, a measure of these economic(hedging) costs---in the sense of a smaller upside potential---can be seen asthe difference between the expected return of both investment vehicles.18

Regulatory Framework of Money-Back Guaranteesfor IPA: The Case of Mutual FundsThe money-back guarantee as described in the German Certification Act canbe represented as a fixed liability of the provider, when it issues the IPA. Ifthe cash value of the financial assets backing the liabilities at the beginningof the retirement phase is lower than the sum of the contributions paidinto the policy, the provider must fill the gap with equity capital. From thispoint of view, it is clear that the money-back guarantee should be subject tosolvency regulation.

Saving products offered by commercial banks (e.g. saving accounts) orinsurance companies (e.g. life insurance products) in Germany are usuallydesigned (at least in part) with fixed interest rates. Nevertheless such isnot the case for mutual funds. The fundamental idea of a collective invest-ment scheme such as a mutual fund is to collect money from many privateinvestors via the offering of fund units, and to invest this money in a well-diversified portfolio of stocks, bonds, and/or real estate. The units of themutual fund are liquid in the sense that they are traded on an active sec-ondary market (e.g. for so-called exchange-traded-funds) or investors canask for redemption of their holdings at net-asset value prices, at any point intime. The investment management company usually assumes no obligationother than that of investing the funds in a reasonable and prudent man-ner, solely in the interest of the investors. It provides no guarantees withrespect to a rate of investment return. Hence, the investor bears all capitalmarket risk and receives the full reward of the financial asset that backs themutual fund units. Because the balance sheets of mutual fund providersare not exposed to financial market fluctuations, they are excluded fromrisk-based solvency capital regulation requirements in Germany, in contrastto insurance companies and commercial banks.19

By contrast, if the provider of an IPA is an investment managementcompany which uses its own mutual funds, the German Federal BankingSupervisory Authority (BAKred)20 requires (conditional) solvency capitalbecause of the statutory ‘‘money back’’ guarantee. This solvency require-ment, published in December 2001, can be modeled in the following way.Let Vt denote the cash value of an IPA at time t , and let Pt be the sum ofthe contributions (including all extra payments by the government) paid

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into the policy until time t . Furthermore, let rf (t , T ) =: rf ,t be the yield attime t on a zero coupon bond maturing at time T (i.e. the planned ageof retirement), taken from the current term structure of German interestrates.

For each IPA, the investment management company must build solvencycapital equal to 8 percent of the total contributions paid into the plan,21

in each period t in which the risk-adjusted cash value of the policy is lowerthan the present value of the contribution:

Vtexp(2.33σ)

≤ Pt(1 + rf ,t )T −t−1 . (9.5)

In this formula,22 σ stands for the monthly volatility of the mutual fundunits backing the pension account. The volatility must be estimated fromhistorical time series returns of the fund unit prices using a window between2 and 5 years. If the policy consists of more than one type of mutual fund(e.g. equity and bond funds), σ is computed as the weighted sum of theindividual fund volatilities according to the asset allocation of the policy.

The economic rationale behind this formula is as follows. At every pointin time, the IPA issuer has the safe investment alternative of investing somepart of the contributions in zero bonds, so that at the end of the investmentperiod at time T the proceeds would equal the participant’s contributionsduring the accumulation phase. The necessary amount to meet the totalcontribution guarantee of Pt at time t is Pt/(1 + rf ,t )T −t , which is the righthand side of formula (9.5). If the provider does not use zero bonds, butinstead employs only stocks to back the IPA, nothing happens as long as thecash value of the policy is ‘‘substantially’’ higher than the present value of thecontributions. ‘‘Substantially’’ higher means, under the German solvencyrule, that given a current cash value of Vt there is a probability of only1 percent (note 2.33 is the 99 percent quantile of the standard normaldistribution) that the uncertain cash value of the policy one month laterVt+1 is lower than the present value of the contributions. This explains therisk adjustment on the left hand side of the solvency formula.

Hence, without capital requirements, an underfunding of the principalliability during the accumulation period is possible. The amount to whichsuch an underfunding is allowed depends on the volatility of the pensionassets and the time remaining to the end of the accumulation period. Forexample (see Table 9-1), if the monthly returns of the pension assets have avolatility of 7.22 percent per month, which when annualized is about 25 per-cent per year (a typical value for German stock funds), the risk-free interestrate is 4 percent p.a., and the remaining accumulation period is 30 years(360 months), then the critical level is only 35.8 percent. This means thatas long as the cash value of the policy exceeds 35.8 percent of the contri-bution paid into the plan, no risk-based-solvency capital is necessary. If the

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TABLE 9-1 Critical Level of Under Funding (as Percent of Contributions) withrespect to the Solvency Formula (9.5)

Volatility (% per month)

End of plan (Years) 0.29 0.58 0.87 1.15 1.44 2.89 5.77 7.2230 30.5 30.7 30.9 31.1 31.3 32.4 34.6 35.825 37.2 37.5 37.7 38.0 38.2 39.5 42.3 43.720 45.4 45.8 46.1 46.4 46.7 48.3 51.6 53.415 55.5 55.9 56.2 56.6 57.0 59.0 63.1 65.210 67.8 68.2 68.7 69.1 69.6 72.0 77.0 79.65 82.7 83.3 83.8 84.4 85.0 87.9 94.0 97.23 89.6 90.2 90.8 91.4 92.0 95.2 101.8 105.32 93.3 93.9 94.5 95.2 95.8 99.1 106.0 109.61 97.1 97.7 98.4 99.0 99.7 103.1 110.3 114.1

Source: Authors’ computations.

time to retirement is only 5 years (60 months), the critical level increasesto 97.2 percent. However, the provider has the possibility of reducing thevolatility of the IPA and the possible amount of underfunding by investingmore of the pension assets in low volatility assets such as bonds.

In summary, with an appropriate asset allocation and depending on theage of the participant, it is possible for the provider of mutual fund-based IPAto avoid capital requirements without jeopardizing the credibility of the prin-cipal guarantee. However, the burden of such a conditional solvency systemis the implementation of an efficient risk monitoring system for each IPA.

Hedging Costs and Hedging Effectiveness ofMutual Fund ProductsIn view of our results concerning the long-run risks of pure stock invest-ments, and given the regulatory environment placing a significant capitalcharge on a fund with too much shortfall risk, it is clear that a sensiblestrategy for a mutual fund must contain some element of risk managementor hedging. As mentioned above, the problem is to provide sufficient credib-ility for promised payments (hedging effectiveness), while at the same timereducing the upside potential of the investment as little as possible, to keephedging costs low. Note that the term ‘‘hedging costs’’ refers neither to theregulatory capital the mutual fund company has to provide, nor to potentialexpenditures for the purchase of derivative contracts. The only source ofhedging costs for the products considered below is a reduction in averageexpected wealth or, equivalently, in the total return on the contributionspaid into the IPA.

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Because of the substantial positive correlation of the financial assets back-ing the pension plans, an IPA provider cannot manage the risk resultingfrom guarantees by using traditional insurance pooling techniques.23 Henceit is necessary to manage underlying risk of the principal guarantee for eachIPA individually.

MethodologyFocusing on products currently offered by German mutual fund companies,we compare them to the simple strategies of investing in stocks or bondsexclusively. In total, we analyze five strategies with respect to their long-runrisk-return profile.

Pure Stock Strategy

This strategy was discussed above with respect to its long-run risks. Giventhe parameter values used in our simulation study, this strategy is likely toproduce the highest expected wealth at the end of the investment period.On the other hand, this strategy can be quite costly for the mutual fundcompany if it must put up substantial solvency capital to render credible itspayment promises.

Pure Bond Strategy

A pure bond strategy follows the opposite approach. To reduce the risk offalling short of the promised wealth at the end of the accumulation period,this strategy invests only in bonds or broadly diversified government bondportfolios. One might expect that this reduces or even completely eliminatesthe shortfall risk, but this benefit also comes at the cost of lower expectedreturns.

Static Portfolio Strategy

This strategy is a mixture of the pure bond and the pure stock strategy. Theportfolio remains unchanged over the whole period, and it contains bothstocks and bonds from the start. With reference to the typical asset allocationof German retirement funds (AS-Funds),24 our simulations for the 15-yearhorizon use an equally weighted stock and bond portfolio, whereas for the30-year investment period we use 75 percent stocks and 25 percent bonds.

Life Cycle Strategy

Popular advice often given to investors is to alter the portfolio compositionwith age. People are usually advised to hold a larger share of the portfolio instocks when young, and then to shift into bonds later on. The idea behindthis strategy is that it would be hard to compensate unfavorable movementson the stock market occurring late in the accumulation period, since little

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time is left, so that this type of risk could be avoided by investing in bonds.The life cycle strategy is an unconditional ‘‘hedge’’ in the sense that morevolatile return opportunities are generally considered too dangerous late inthe investment period, irrespective of the performance of stocks before therebalancing date. We implement this strategy by defining fixed points in timeat which the portfolio composition is changed, with more and more weighton bonds instead of stocks. The exact dates and compositions are as follows:For an investment horizon of 15 years, the plan is assumed to start with40 percent of the allocations going into equity and 60 percent into bonds.After 5 years this allocation changes, and for the remaining time 10 percentgo into stocks and 90 percent into bonds. In the case of a 30-year plan, thereis an initial period of 10 years with pure stock investment, followed by 5 yearswith an allocation of 70 percent equity and 30 percent bonds. After another5 years, the allocation of the contributions is again changed to 40 percentequity and 60 percent bonds. Over the remaining 10 years, 90 percent ofthe contributions go into bond funds and the remaining 10 percent intostocks.

Conditional Hedging Strategy

This strategy aims at combining the performance advantage of a pure stockstrategy with the risk-reducing effect of a pure bond strategy. As opposedto the life cycle strategy, however, the decision to shift from one investmentinto the other is not driven by an exogenous variable like age, but ratherby the performance of the respective investments. For this reason, such astrategy represents a ‘‘conditional hedging’’ approach. Usually one startsout with a pure stock investment and shifts to bonds as soon as a certaincritical level of wealth is reached. In this case, subsequent contributions goin bonds until the safety level is again exceeded, when the strategy switchesback to a 100 percent stock investment. An important parameter for thistype of strategy is the critical level of wealth at which the investment rule (forsubsequent contributions) changes. To link this critical value to the inter-vention line set by the regulatory authorities in Germany (see equation 9.5),we set the critical level of wealth (as an example) to 75 percent above theintervention value defined according to the solvency equation (9.5).

A possibility not discussed up to now is the use of derivative assets to pro-tect the value of an investment plan against shortfall risk. The appropriateinstrument here would be a put option on the value of the plan, with a strikeprice equal to the sum of the nominal payments. However, the applicationof put options in this context is not without problems. First of all, due to thevery long maturity of the savings plans, any option would be very expensive,and the cost would have to be paid up front (at the beginning of the accumu-lation period) which raises financing questions. Second, it seems unlikelythat a put with such a long time to maturity would be offered at all, so that a

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roll-over strategy would become necessary with all the risks involved in termsof prices and liquidity. Third, for the put option to be of real value to theinstitution holding it, the seller would have to demonstrate that it could actu-ally cover its liabilities at the end of the accumulation period. In practice,there would always be doubts concerning the actual risk-reduction poten-tial of such an option. Finally, there is a significant operational problemin using put options, since all the accounts have to be protected individu-ally. This means that for every IPA, the provider would have to hold a putoption with the appropriate strike price and time to maturity. This seems toocostly and complicated for the typical institution, so that hedging strategiesusing ‘‘physical’’ financial derivatives will not be considered further in thefollowing analysis.

We analyze the five strategies described above in terms of wealth levels(or total returns) and required regulatory capital. Since there are no closed-form expressions for the statistics of interest, we use Monte Carlo simulationto generate a large number of paths for the evolution of the savings plans.The relevant statistics for total returns (relative to a benchmark) and regu-latory capital are then evaluated on the basis of these scenarios. The keyingredient in such a simulation is a suitable model to describe the dynamicsof the relevant funds and the short rate of interest. For the funds we usethe standard capital market model, representing asset price movementsby means of correlated Wiener processes. The dynamics of the short rateare given by the one-factor model suggested by Cox, Ingersoll, and Ross(1985). While we assume constant correlations between the risk factors, thecovariances will vary due to the fact that the conditional standard deviationfor the short rate will in general not be equal to the unconditional value.

The time series used to estimate the process parameters (mean returns,volatilities, correlations) are the monthly log returns of the German stockindex DAX representing the stock index fund, the log returns of the bondperformance index REXP as the bond index fund as well as the 1-yearinterest rate as a proxy for the short rate. Parameters were estimated viaa maximum-likelihood approach, the estimates are presented in Tables 9-2and 9-3. As discussed above we subtracted the equivalent of 0.5 percentp.a. from the monthly average return on the investments to take potentialadministration costs into account.

The CIR process is very popular in interest rate modeling. This is mainlydue to the fact that it is able to generate both mean-reversion in interestrates as well as non-negative rates with probability one. Since the processexhibits mean-reversion, the sign of the drift component (i.e. the expectedchange in the short rate over the next time interval) depends on whetherthe process is currently above or below its long-run mean. How quicklythe process reverts back to this long-run mean is determined by the speedof mean-reversion. In Table9-3, κ (kappa) represents this speed of meanreversion, θ (theta) stands for the long-run mean of the interest rate, while

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TABLE 9-2 Descriptive Statistics for Risk Factors in Germanyfrom January, 1973 to December, 2001

Asset Mean (% p.m.) Volatility (% p.m.) Correlations

Stocks Bonds

Stocks 0.7967 5.5800 1 0.2051Bonds 0.5683 1.1200 1


TABLE 9-3 Descriptive Statistics for the German Short RateProcess from January, 1973 to December, 2001

κ θ σ Correlations Innovations with

Stock Returns Bond Returns

0.1494 0.0539 0.0511 0.1417 −0.7009

The process estimated is the CIR process drt = κ(θ −rt )dt +σ√rt dWt ,where κ is the speed of mean-reversion, θ is the long-run mean of theshort rate, and σ is the volatility of changes in the short rate. dWt isthe increment of a standard Wiener process.


σ (sigma) denotes the volatility of changes in the short rate. The marketprice of interest rate risk was set equal to zero for reasons of simplicity.

To ensure stability of the simulation results, we base our analysison 3,000,000 simulations for each of the respective strategies. We thencompute:

(1) statistics related to hedging costs: the mean of the total return gen-erated by the respective strategies for the different points in time(months);

(2) statistics related to hedging effectiveness: the shortfall risk and therequired regulatory capital for the respective strategies.

The model assumes equal contributions into the plan occurring at thebeginning of each month, and a front-end load of 5 percent proportionalto the unit price for the stock fund and 3 percent for the bond fund. Theseloads are comparable to the current German mutual fund fee structure.Vi ,t denotes the uncertain total wealth of the IPA in month t(t = 1, . . . , T )in simulation run i(i = 1, . . . , n), and zi ,t represents the critical level ofwealth in month t according to formula (9.5) determined by the BAKred.25

If Pt represents the sum of payments into the plan until time t , the average

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compounded (multiyear) return (EWt ) of the IPA at the end of month t isgiven by:

EWt = 1nn∑

i=1

Vi ,t − PtPt

. (9.6)

The probability of a solvency capital charge (CPt ) in month t is estimated by:

CPt = 1nn∑

i=1max[zi ,t − Vi ,t ]0I(−∞,zi ,t )(Vi ,t ), (9.7)

where the indicator variable I(a,b)(X ) is equal to one if X ∈ (a, b) and zerootherwise. The mean solvency capital charge (MCt ) at time t month afterthe beginning of the plan normalized by the sum of the contributions Ptpaid into the IPA, is given by

MCt = 1nn∑

i=1

Ci ,tPt

. (9.8)

According to the regulatory authorities, the solvency capital charge Ci ,tdepends on how far the mutual funds based IPA wealth falls short of thecritical level. The rule says that the capital charge is at least 8 percent whenwealth falls below the critical level. If the amount of the shortfall exceeds8 percent, the capital charge is increased accordingly to cover the gap.Hence Ci ,t must to be calculated according to the following formula:

Ci ,t =

0.08 · Pt 0 < 1 − Vi ,t/zi ,t ≤ 0.08(1 − Vi ,t/zi ,t ) · Pt 1 − Vi ,t/zi ,t > 0.080 1 − Vi ,t/zi ,t < 0

. (9.9)

The mean conditional capital charge (MCCt ) at month t given that a capitalcharge has occured is computed according to:

MCCt = MCt/CPt. (9.10)

ResultsOur results for the expected total return of savings plans based on the dif-ferent investment strategies are given in Table 9-4. It is no surprise that thepure stock strategy does best in terms of this measure, since stocks have thehighest expected monthly return. This also causes the differences betweenthe respective strategies to increase with time. Nevertheless, it is interestingto note how close the conditional hedge strategy comes in terms of expectedtotal return. Even after 30 years, the difference to the pure stock strategyis only about 3.5 percent of the contributions paid. This can be taken as a

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TABLE 9-4 Expected Total Return in Germany (in % of contributions)

Year 1 5 10 15 30

100% bond0.80 16.26 40.17 70.67 225.38

(0.80) (16.26) (40.17) (70.67) (---)

100% stock1.38 29.09 78.78 154.06 731.60

(1.38) (29.09) (78.78) (154.06) (---)

Static1.72 26.31 68.79 130.38 554.59

(1.08) (22.44) (58.03) (107.36) (---)

Life cycle1.38 29.09 78.78 140.13 384.93

(1.03) (21.39) (46.73) (81.36) (---)

Conditional hedge1.38 29.09 78.77 153.93 728.06

(1.33) (26.07) (67.40) (126.54) (---)

The table gives the expected total compounded return for the different IPAplans at different points in time for a 30-year accumulation period (numbers inparentheses are for an accumulation period of 15 years). For example, an entryof 16.26 for the 100 percent bond strategy in year 5 means that the value of an IPAwith an investment horizon of 30 years is after 5 years on average 16.26 percenthigher than the sum of the contributions over the first 5 years.


first indication that this type of strategy might be an interesting compromisebetween the return potential of a pure stock strategy and the risk-avoidingproperty of a pure bond approach.

Nevertheless, expected wealth is just one measure to be considered; anysensible comparison of the given products must also focus on risk measures.The risk of the different strategies is measured by the regulatory capitalcharge that the mutual fund company adopting these strategies would face.Table 9-5 indicates that, for an investment horizon of 30 years, no regu-latory capital is needed over the first 5 years for any of the strategies. Thepure bond strategy can even be regarded as entirely risk-free with respectto regulatory capital charges. The life cycle approach also exhibits very lowcapital charges on average. This strategy seems to be an interesting altern-ative to a pure bond investment, given its advantage in terms of expectedreturn.

As expected, the pure stock strategy balances its high return potentialwith an ‘‘expensive’’ regulatory capital level. It requires more than threetimes the regulatory capital than the conditional hedge strategy, whichis in second place with respect to this criterion. Furthermore, Table 9-5provides insight into the impact of the investment horizon. Long-termstrategies generally exhibit lower risk than the 15-year plans with theonly exception being the life cycle strategy. The fundamental reason forlonger-term strategies requiring less regulatory capital than shorter-term

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TABLE 9-5 Mean Regulatory Capital Charge in Germany (as %of Contributions)

Strategy Year

1 5 10 15 30

100% bond0 0 0 0 0

(0) (0) (0) (0) (0)

100% stock0 0

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TABLE 9-6 Probability of a Regulatory Capital Charge inGermany (in %)

Strategy Year

1 5 10 15 30

100% bond0 0 0 0 0

(0) (0) (0) (0) (0)

100% stock0 0

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TABLE 9-7 Mean Conditional Regulatory Capital Charge in Germany (as% of Contributions)

Strategy Year

1 5 10 15 30

Pure bondn.def. n.def. n.def. n.def. n.def.

(n.def.) (n.def.) (n.def.) (n.def.) (n.def.)

Pure stockn.def. n.def. 9.46 11.59 19.90

(n.def.) (9.92) (14.31) (18.63) (---)

Staticn.def. n.def. n.def. 9.31 14.71

(n.def.) (n.def.) (8.75) (10.17) (---)

Life-cyclen.def. n.def. 9.46 8.04 n.def.

(n.def.) (n.def.) (n.def.) (8.00) (---)

Conditional Hedgen.def. n.def. 8.00 9.00 13.29

(n.def.) (8.06) (8.90) (9.95) (---)

The table gives the average conditional regulatory capital that has to be put up forthe different IPA plans at different points in time for a 30 year accumulation period(numbers in parentheses are for an accumulation period of 15 years), that is, theaverage amount of regulatory capital that is necessary, given that regulatory capitalhas to be put up at all. For example, an entry of 9.00 for the conditional hedge strategyin year 15 means that in this year on average 9.00 percent of the sum of contributionsover the first 15 years had to be provided as regulatory capital in those cases wherecapital had to be put up at all. Note that this number is not defined (‘n.def.’), if theempirical probability of having to provide regulatory capital is equal to zero.


higher wealth over the whole investment period and the average conditionalregulatory capital is also lower. So if one were to compare the differentproducts on the basis of these two measures only, the static strategy wouldbe dominated. Note, however, that the probability of a capital charge islower for the static strategy.

The analyses thus far focus on the statistical output, but it is also import-ant to assess the administrative costs generated by each plan. Cost will notbe major for the strategy products such as the pure bond, the pure stockplans, the static, and the life cycle strategies. However, for the conditionalhedge, the need to shift incoming distributions across asset classes depend-ing on how much wealth has been accumulated in the plan, might implyconsiderable administrative effort. It is therefore of interest to examine therelative frequency of shifts from one asset class to the other, when a condi-tional hedge strategy is run. Here again, time is an important factor, sincefor the 15-year plan the mutual fund company must change the asset classin 98 percent of the paths generated by the simulation, whereas this needarises in only 26 percent of the cases for the longer horizon. In any case,costs must be taken into account when strategies are compared with respectto their practical application.

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In summary, it is not possible to identify the overall dominating investmentproduct or strategy. With a few exceptions, higher potential in terms ofaverage wealth usually comes at the cost of higher regulatory capital. It isimportant in this context to look at the average amount of regulatory capitalconditional on the event that capital actually has to be put aside. Here itbecomes obvious that strategies with a fixed stock investment can producesignificant risks. This risk is mitigated when the conditional hedge strategy isemployed. Nevertheless, additional administrative cost must be considered.

ConclusionsDue to the severe financing problems of standard pay-as-you-go pension sys-tems in many countries, alternative vehicles for retirement financing have tobe developed. In Germany, such a new system was installed when the GermanRetirement Saving Act was passed by the legislative body. The governmentoffers significant tax relief for investment products meeting certain require-ments, the most important of these being a guarantee promising that thecash value of the IPA at the end of the accumulation period will be at leastas high as the nominal sum of the contributions. To lend sufficient cred-ibility to the payment promises made by institutions providing investmentproducts for these savings plans, the regulatory authorities in Germany haveimposed a capital charge in case the value of the savings plan falls below acertain critical level.

At first sight, it seems that in order to implement such a principal guar-antee, complicated and expensive financial products like derivatives areneeded. However, as we have shown, there are other ways of achieving asometimes practically risk-free position without using options or similarinstruments. We analyze in detail various strategies aimed at combiningthe potentially return-increasing properties of equity investment with therisk-reducing characteristics of bond investments. These strategies offer areal-world application of the tools and methods of capital market theory.Of course, the trade off between return and risk is always at the core ofthe analysis. Yet in the context of this chapter it is important to recognizethat variance is not the most important measure of risk. As opposed to amore traditional approach we consider shortfall and the need of regulatorycapital as the two most important types of risk.

The strategies analyzed here range from simple pure bond or equityinvestments, to mixed equity-bond funds and products offering a changein portfolio composition at pre-defined points in time, to highly sophistic-ated products with conditionally changing investment styles. One of the keyresults of our study is that these dynamic strategies, switching from stocksinto bonds whenever the value of the savings plan falls below some criticalsolvency ratio set by regulatory authorities, perform rather well in terms ofexpected total returns for long investment horizons. They come close to

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pure stock investments with respect to the average value they generate forthe investor. However, it is also very important for the financial institutionto keep an eye on the expected amount of regulatory capital required bya certain investment strategy. Due to the conditional change in allocationwhen the critical regulatory value is reached, the expected capital charge issignificantly smaller than in the case of a pure equity investment.

Besides the basic type of strategy, the length of the investment horizonis an important factor for the risks and rewards of alternative strategies.In general, the longer the maturity of the plan, the lower the expectedcapital charge, since the critical level set by the authorities in Germany con-tains a discount factor, the higher the expected total return. Nevertheless,it is important to consider other risk variables as well in this. We are farfrom claiming that one of the strategies discussed here should be seen asuniformly superior to any other. Rather we seek to point out the benefitsand risks offered by the different types of products, to provide a basis for athorough discussion of the issues involved in product design and regulation.

Appendix

Derivation of the Solvency FormulaConsider an investment plan where payments into an IPA are made at equallyspaced points in time t = 0, 1, . . . , T (e.g. months). Let Pt denote the sumof payments up to time t , T the planned terminal date of the plan (equalto the beginning of the payout phase), and q(rf ,t , T − t) = (1 + rf ,t )t−T thediscount factor with risk-free rate rf ,t and remaining time to maturity T − t .Without loss of generality we assume that the investor holds exactly oneshare of the fund at time t . We are interested in the solvency ratio Vt/Pt attime t , which makes sure that the uncertain market value of the shares Vt+1at time t + 1 is less than the sum of payments Pt into the plan discountedup to time T , that is, less than Pt · q(rf ,t , T − t − 1), with a probability of atmost ε.

To be able to quantify this shortfall risk, we have to specify a model forthe random evolution of the value of the investment shares. Here we makethe standard assumption that the dynamics of this value can be describedby a geometric Brownian motion. This implies that the relative change invalue (i.e. the log-return) ln(Vt+1) − ln(Vt ) is normally distributed withmean µ und variance σ 2. Formally we obtain the desired solvency ratioas the solution of the following inequality:

Prob[Vt+1 < Pt · q(rf ,t , T − t − 1)]= Prob[ln(Vt+1) < ln(Pt · q(rf ,t , T − t − 1))] ≤ ε (9.A1)

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Using the above distributional assumption inequality (9.A1) is equivalent to

ln(Vt ) + µ ≥ ln(Pt ) + ln[q(rf ,t , T − t − 1)] + N1−ε · σ , (9.A2)where N1−ε is the (1 − ε)-quantile of the cumulative standard normaldistribution. Under the additional (conservative) assumption26 that theone-period expected return is equal to zero (i.e µ = 0) inequality (9.A2)can be written as

Vt/ exp[N1−ε · σ ] ≥ Pt · q(rf ,t , T − t − 1). (9.A3)Setting N1−ε = 2.33, which implies a tolerated shortfall probability of notmore than 1 percent, this represents the equation (9.5) for the solvencyratio presented in the main text.

Notes1This Act is also known as ‘‘Riester Reform.’’ Walter Riester was the German LaborMinister responsible for the reform of the pension system in the year 2001. Moreformally, the reform as a whole alters several existing laws including (among others)the social security law, the income tax law, the occupational pensions law, the socialwelfare law, the civil law, the law governing investment management companies, andthe law governing insurance companies.2Similar to that, the maximum pension for civil servants is being reduced from75% to 71.75% of the last salary.3In addition, the government also promotes the ‘‘second pillar’’ occupationalpension system, for example, by establishing a new funding vehicle called ‘‘Pen-sionsfonds.’’4An exemption is, that a part of the pension plan (min¤ 10,000 and max.¤ 50,000)can be withdrawn during the accumulation phase to finance own house. This amountmust be paid back (at a zero interest rate) into the IPA before the beginning of thedistribution phase.5In the case of a life annuity the provider must promise lifelong constant or increas-ing (monthly) payments to the annuitant. In the case of a capital withdrawal plan(typically offered by mutual fund and/or bank providers) at least 60% of the accu-mulated assets (but not less than the contributions paid into the IPA) must be usedfor constant or rising periodic payments. At latest at the age of 85 the balance mustrevert into a life annuity, whereas the benefits cannot be less than the last paymentreceived before that age. In addition, not more than 40% of the accumulated assetscan be used for a withdrawal plan with variable pension payments (reflecting thereturn of a specific asset portfolio).6Especially for traditional life insurance policies, it is conventional (until now)that distribution costs are charged as front end loads on the first premiums (viathe so-called zillmer-adjustment) resulting in no or low early cash values for thepolicyholder.7An exemption are financial derivatives (e.g. option, futures, swaps), which can beused within an IPA for hedging purposes only.8Not more than 15% of total contributions can be deducted from the principalguarantee level, if the IPA include insurance coverage against disability. In the case

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of a switch to a new provider during the accumulation phase, the policyholder getsfrom the new provider a guarantee on the policy’s cash value at the time of transferplus new premiums.9See also Lachance and Mitchell (Chapter 8, this volume).10For surveys cf. Albrecht, Maurer, and Ruckpaul (2001) and Kritzman and Rich(1998).11Cf. Bodie (2001) and Bodie (2002) for a critique of these simple arguments.12The concept of shortfall risk was introduced in finance by Roy (1952) and Kataoka(1963), expanded and theoretically justified by Bawa (1975), and Fishburn (1977,1982, 1984). It is widely applied to investment asset allocation by Leibowitz, Bader,and Kogelman (1996) and used by Albrecht, Maurer, and Ruckpaul (2001), Asness(1996), Butler and Domian (1991), Leibowitz and Krasker (1988) and Zimmermann(1991) to judge the long term risk of stocks and bonds.13The MEL is closely connected with the tail conditional expectation, which is givenby TCE = E(R | R < z) = z − MEL. The TCE has some favourable features,for example, it is (in contrast to the shortfall probability) a coherent risk meas-ure with respect to the axioms developed by Artzner et al. (1999). In additionthe MEL is a suitable version of the mean excess---respectively mean excess loss-function E(X − z | X > z) considered in extreme value theory. For extreme-valuemethods in financial risk management cf. Borkovic and Klüppelberg (2000) andEmbrechts, Resnick, and Samorodnitsky (1999). Very early, Gürtler (1929) intro-duced the MEL as ‘‘Mathematisches Risiko’’ to evaluate the underwriting risk ofinsurance companies.14The Black and Scholes formula follows directly only for a single lump sum pen-sion payment and not for a series of contributions; see also Lachance and Mitchell(Chapter 8, this volume).15As shown by Albrecht, Maurer, and Ruckpaul (2001) the risk measures can bederived analytically in the case of a lump-sum investment.16Feldstein, Ranguelova, and Samwick (2001: 60) use a similar procedure to accountfor potential administration costs.17The large number of simulation paths is necessary to receive a precise picture ofthe worst case risk measure MEL, especially when the shortfall probability is low.18Despite the fact that the expected return is the most common measure of the‘‘reward,’’ ‘‘return,’’ or ‘‘value’’ of financial investments it is---especially in a downsiderisk context---possible to measure the upside potential more directly, c.f. Holthausen(1981) or Albrecht, Maurer, and Möller (1998). AQ:

Albrecht etal (1998)not listedpls chk

19According to German Investment Company Law (KAGG), the minimum equitycapital for investment fund management companies (i.e. the provider of the pensionproducts) is ¤ 2.5 Millions.20Since May of 2002 the BAKred is a Department of the new German FederalFinancial Supervisory Agency.21In addition to the solvency equity capital, investment management companiesmust build supplementary reserves, if the difference between the present value ofthe contributions and the risk adjusted cash value of the policy exceeds 8% of thetotal contributions.22The formula is explained in more detail in Appendix B.23See also Lachance and Mitchell (Chapter 8, this volume).

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24AS-Funds (Altersvorsorge-Sondervermögen) are special mutual fund productsregulated in the Investment Management Company Act, which the German gov-ernment introduced in 1998 for retirement saving. In contrast to usual balancedfunds, AS-Funds can invest into real estate, require a saving plan of at least 18 years,and are subject to some quantitative investment restrictions. For more details seeLaux and Siebel (1999).25C.f. BAKred (2001).26This assumption is indeed conservative, since it increases the shortfall probabilitycompared to the common case µ > 0. Furthermore it is no longer necessary toestimate expected returns (e.g. from historical time series), which are subject tomuch larger estimation risks than volatilities. See, for example, Merton (1980).

References* Albrecht, Peter, Raimond Maurer, and Matthias Möller. 1999. ‘‘Shortfall-

Risiko/Excess-Chance-Entscheidungs-Kalküle: Grundlagen und Beziehungenzum Bernoulli-Prinzip.’’ Zeitschrift für Wirtschafts- und Sozialwissenschaften 118:249--274.

------ ------ and Ulla Ruckpaul. 2001. ‘‘The Shortfall-Risk of Stocks in the Long Run.’’Journal of Financial Market and Portfolio Management 4: 427--439.

* Ammann, Manuel and Heinz Zimmermann. 2000. ‘‘Evaluating the Long-Term Riskof Equity Investments in a Portfolio Insurance Framework.’’ Geneva Papers on Riskand Insurance 25: 424--438.

Artzner, Philippe, Fredy Delbaen, Jean-Mar Eber, and David Heath. 1999. ‘‘CoherentMeasures of Risk.’’ Mathematical Finance 9: 203--38.

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Bawa, Vijay S. 1975. ‘‘Safety First, Stochastic Dominance and Optimal PortfolioChoice.’’ Journal of Financial and Quantitative Analysis 13: 255--271.

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* Bernstein, Peter L. 1996. ‘‘Are Stocks the Best Place to be in the Long Run?A Contrary Opinion.’’ Journal of Investing 5: 9--12.

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